'You just want them to be happy' … Suzi Gomez-Pizzo, left, assists Natalie Dessay, singing Cleopatra in Giulio Cesare at the Met in New York. Photograph: Mike McGregor for the Guardian Mike McGregor/Guardian

"Much of mankind is divided into two categories: the enablers and the enabled." That's what the late critic Roger Ebert wrote after watching the 1983 film The Dresser, about an ageing actor and his devoted assistant. And backstage at an opera, that division couldn't be more stark. Singers whose stardom is stratospheric are each assigned their own dresser – a personal assistant who is often more like an indulgent parent.

It is the dresser who will hover in the wings with a hot towel for whenever their charge wants to inhale fresh steam; it is the dresser who will dash off to find the right bar of chocolate should the diva get a craving halfway through act two; and it is the dresser who will murmur just the right words of encouragement as their star nervously waits in the wings before singing that first aria.

Suzi Gomez-Pizzo has been doing this job for 30 years. One wall of her office, at New York's Metropolitan Opera, is devoted to baby photos – as the buckler of belts and the lacer of corsets, she's always the first to know when a soprano is pregnant. She wears glasses, a ponytail and a utility waistcoat into which all her equipment is crammed. "Yeah, I'm fully laden," she says. "Needles, thread, pliers, scissors ..."

"Show them your lamp!" says French soprano Natalie Dessay, who likes to tease her. Gomez-Pizzo flashes a miner's light on and off. "Yeah," she says. "It helps me to see in the dark."

"She doesn't have the helmet though," Dessay adds, with a sly smile.

Gomez-Pizzo, who calls Dessay mi amor, explains that the Frenchwoman was the first singer assigned to her when she started working at the Met seven years ago. Tonight, Dessay is singing Cleopatra in Giulio Cesare, the Handel opera based on the meeting of Julius Caesar and the Egyptian queen in 48BC; this Saturday's performance will be screened live to cinemas around the world, including the UK.

We're in Dessay's dressing room, where Gomez-Pizzo is making the final tweaks to the emerald green pantaloons Cleopatra makes her entrance in. This is the first of eight outfits tonight, many requiring rapid changes in a booth in the wings. Dessay, petite and intense, glares at herself in the mirror. I ask what she needs in a dresser. "Calm," she says. "Very calm and professional. When you do the changes, she's like: 'Do this, take this, put this on, don't move.'" She demonstrates with hand movements. "She's very professional, like a surgeon."

"Saaaaame difference!" says Gomez-Pizzo, who turns to me and adds: "You just want them happy."

With slipper straps blackened to her satisfaction, Dessay stands with her hands on hips, then widens her heavily kohled eyes as she examines herself in the mirror. It's a gesture of readiness. As the words "Cleopatra to stage" come over the speakers, Gomez-Pizzo puts a hand on the small of Dessay's back and guides her out. The two swoop from her dressing room towards the vast backstage area and the wings.

This is a darkened world of contained hysteria, where eyes watch monitors and instructions are whispered into headsets. Dessay is about to step out into another world – full of bright lights and an audience of 3,500 people. "I usually just give her this moment," Gomez-Pizzo whispers to me without taking her eyes off Dessay, who is standing a few feet away with her head bowed, taking long, slow breaths.

And then she's on. For the next four and a half hours (including two half-hour intervals), she and countertenor David Daniels dance, act and sing one of Handel's most technically demanding works. As the night progresses, the dressers idle around, chatting and browsing on iPads. Among them is Lou Valantasis, a laconic 46-year-old in black glasses and lime-green trainers who's the head of men's wardrobe. As such, he explains, "I'm a dresser to many …"

"But none as fun as me!" interjects Daniels, generally regarded as the world's best countertenor. Daniels is singing the title role and it's a marathon: he has eight arias, which means he relies heavily on Valantasis. "I need him to be supportive and fun and he is. He's incredibly chill." A little later, as if to test this, Daniels shrugs off his coat and flings it to the ground. Unhurried, Valantasis retrieves it, saying dryly: "Next act he drops his pants."

Valantasis finds he has to be part cheerleader, part coach and part confidant. "You listen," he says. "I'm not necessarily a conversationalist and I'm not trying to figure them out. But, as I'm standing there putting them into something, they say things. They're fearing for their health, the weather, the run – all sorts of things. You learn what they want. Once in a while, there's a panic. You have to throw yourself into it and say, 'It's you and me, babe, we're gonna do this thing.'"

For much of the performance, Gomez-Pizzo perches on some steps in the wings, half in darkness, and looks out on to the stage holding a huge bottle of water ready for Dessay. She was once left literally holding the baby, she says, when a soprano's nanny cancelled.

Finally, after Cesare and Cleopatra have declared their love, the cast amass for the closing chorus. The applause is overwhelming. "Yay!" Gomez-Pizzo says. "That's for her! Yay mi amor, my little girl." But as the applause keeps rolling, Gomez-Pizzo rolls her eyes, taps her watch and says, "C'mon diva, mommy wants to go home." Finally, Dessay makes to leave the stage and Gomez-Pizzo, squinting out from the wings, catches a glimpse of her face. "Oh, she's happy – good."

Dessay sweeps through applauding stagehands who part before her. And Gomez-Pizzo follows behind, arms cradling the train of her dress.